3D Thermal Simulation Of Resistive Heating

Joule heating, also known as resistive or Ohmic heating, is the power lost to heat as electrical current flows down a conductor. We were introduced to Joule’s first law (Power dissipation = I²R, VI, V²/R) way back in high school. From an electronics thermal simulation perspective it requires a full 3D electrical flow simulation to be conducted, and from that the Joule heating power dissipation distribution feeds into the 3D thermal simulation in the usual way.

To model this, electrical boundary conditions are imposed on the periphery of a 3D solid representation of the conductor. The subsequent 3D electro-thermal simulation process solves for current and voltage potential and uses the Joule heating power as a cell-by-cell power source term in the solution of temperature. Typical applications will include busbars, PCB/BGA substrate power and ground planes, leadframes, and of course fuses, anywhere where the resistive heating may play a dominant role in the total power dissipation in the system. This is especially true of high-current power electronics where the thermal dissipation in the power delivery system can often play as important a role as the semiconductor power dissipation.

Here’s a simple and well-recognized example of a fuse, mounted on a simple PCB (Figure 1, fuse housing hidden for clarity). A current value is defined at the trace face leading up to the fuse, a fixed voltage value defined on the edge of the ground plane on the underside of the PCB. A via connects the trace to the ground return.

Figure 1. PCB Mounted Fuse

The resulting Joule heating power dissipation is shown in Figure 2. High power density levels are seen in the fuse section, achieved by design in this application. Being a 3D simulation the power density is shown as power per volume, in this case mW/mm³.

Figure 2. Joule Heating Power Dissipation Distribution

The resulting temperature is the item of most interest. Figure 3 shows the hotter temperatures that exist in the central section of the fuse.

Figure 3. Temperature Variation in the Fuse

Fusing involves a coupling between the electrical and thermal worlds. An increase in temperature will result in an increase in electrical resistivity. That will increase the current density, which will increase the Joule heating power, increase the temperature, and so on. If the heat is removed fast enough, a balance is achieved and things settle down to a constant temperature rise. If the coupling is too strong, specifically under high current conditions, temperature rise will run away until the fuse gets so hot it melts.

Here’s another example showing the effects of Joule heating in a PCB power delivery net, PDN, Figure 4. Resulting plots of voltage potential (all about the same, the PDN is working as expected, providing that voltage potential all over), magnitude of current density, the resulting Joule heating power dissipation and finally the resulting temperature are shown.

Figure 4. Power Delivery Net Electrical and Thermal Distributions

Generally, where there is any necking or reduction in cross-sectional area of the current flow, there is a proportional increase in Joule heating power dissipation and resulting temperature rise. This is especially true in and around vias, also true around corners. If such a simulation indicates that the resulting temperature rise is too large, a redesign can be considered that increases the cross sectional area. This can either be achieved by (a costly) increase in power plane Copper thickness or, more commonly, an increase in the XY size of the plane at that location.

In this case a very small temperature rise above ambient is observed. For ‘typical’ digital electronics it’s the power dissipation in the die of active devices that dominates the thermal behavior of the system, not the Joule heating in the PDNs. For power packages however, as newer versions of packages continue to demonstrate a decrease in Rds(on), the proportion of total power dissipation that comes from Joule heating (in the leadframe) likewise increases. This is leading to an increased focus on 3D Joule heating thermal simulation to ensure accurate package thermal metric determination.